Hom-versions of the Combinatorial Grothendieck Conjecture II: Outer Representations of PIPSC- and NN-type

Abstract

In the present paper, we continue our study, which was initiated in the previous paper of the present series of papers, of combinatorial anabelian geometry of (not necessarily bijective) continuous homomorphisms between PSC-fundamental groups of semi-graphs of anabelioids of PSC-type. In particular, we continue to study certain Hom-versions of the combinatorial versions of the Grothendieck conjecture established in some previous works, i.e., to study certain sufficient conditions of certain group-theoretic compatibility properties described in terms of outer representations. The outer representations we mainly concern in the present paper are of PIPSC-type and of NN-type, both of which are of substantial importance in the study of algebro-geometric anabelian geometry of configuration spaces of hyperbolic curves. We also include, as a preparation for one of the main results, a presentation of a “reduction technique”, namely, a technique of reduction to the “compactified quotients” of (various open subgroups of) the PSC-fundamental groups under consideration, in a similar vein to the previous paper where we included other two “reduction techniques”.